Format and topics for final exam
Math 129a

General information. The final exam will be a comprehensive timed exam, held Fri May 17, 12:15pm-2:30pm, in our usual classroom. The test will cover all of the material on the three in-class midterms, as well as sections 5.3, 5.5, and 6.1-6.2 of the text. No books, notes, calculators, etc., are allowed. Most of the exam will rely on understanding the problem sets (including problems to be done but not to be turned in), the quizzes, and the definitions and theorems that lie behind them. If you can do all of the homework and the quizzes, and you know and understand all of the definitions and the statements of all of the theorems we've studied, you should be in good shape.

You should not spend time memorizing proofs of theorems from the book, but you should defintely spend time memorizing the statements of the important theorems in the text, especially any named theorems.

Types of questions. The final exam will include the usual computations, statements of definitions and theorems, true/false with justification, and paragraph-style questions.

Definitions. The most important definitions and symbols we have covered are:

5.3 diagonalizable D, P notation
5.5 Markov chain states of a Markov chain
transition matrix regular Markov chain
differential equation general solution
initial condition particular solution
6.1 norm (length) of v ||v||
distance between u and v dot product
u·v orthogonal (perpendicular)
orthogonal projection of v onto a line
6.2 orthogonal set orthogonal basis
unit vector orthonormal basis
orthogonal complement Wperp
orthogonal projection of v onto W distance from v to W

Examples. Make sure you understand the following examples.

Sect. 5.3:
Determining if a matrix is diagonalizable (Exs. 3-4, pp. 277-279).
Sect. 5.5:
Finding general and particular solutions to a system of differential equations (pp. 296-297).

Theorems, results, algorithms. The most important theorems, results, and algorithms we have covered are listed below. You should understand all of these results, and you should be able to cite them as needed. You should also be prepared to recite named theorems.

Sect. 5.3:
Diagonalizability Theorem (Thm. 5.2). Test for a Diagonalizable Matrix (p. 277). A matrix with n different eigenvalues is diagonalizable (p. 277). Distinct Eigenvalues Theorem (Thm. 5.3).
Sect. 5.5:
Regular Markov chains Theorem (Thm. 5.4).
Sect. 6.1:
Algebraic properties of dot product and length (Thm. 6.1); Pythagorean Theorem (Thm. 6.2).
Sect. 6.2:
Nonzero orthogonal sets are linearly independent (Thm. 6.5). Gram-Schmidt works (Thm. 6.6). Orthogonal complement is a subspace (p. 327). (Row A)perp=Null A; dimW+dimWperp=n. Orthogonal projection onto W is closest vector in W (p. 331).
The Long Theorems (handout): Fat Matrix, Tall Matrix, Long Theorem.

Types of computations. You should also know how to do the following general types of computations. (Note also that on the actual exam, there will be problems that are not of these types. Nevertheless, it will be helpful to know how to do all these types.)

Sect. 5.3:
Computing Am using D and P (pp. 270-271). Test for a Diagonalizable Matrix (p. 277) (bases of eigenspaces). Algorithm for Matrix Diagonalization (p. 277) (finding D and P).
Sect. 5.5:
Solution to y'=Ay when A is diagonalizable.
Sect. 6.1:
Computing dot products, lengths, distances; are two vectors orthogonal?; orthogonal projection of a vector onto a line.
Sect. 6.2:
Coefficients of a vector in an orthogonal basis (p. 322) and an orthonormal basis (p. 326). Using Gram-Schmidt to get an orthogonal basis. Going from orthogonal to orthonormal basis. Basis for Wperp (pp. 328-329). Orthogonal projection of v onto W (pp. 329-331).

Not on exam. Section 5.5: Harmonic motion, difference equations. Section 6.1: Cauchy-Schwartz inequality, triangle inequality.

Good luck.